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Lesson 9-4

Lesson 9-4. Inscribed Angles. Objective: To recognize and find measures of inscribed angles, and to apply properties of inscribed figures. Inscribed Angle. Inscribed Angle: An angle whose vertex lies on a circle and whose sides are chords of the circle (or one side tangent to the circle).

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Lesson 9-4

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  1. Lesson 9-4 Inscribed Angles Objective: To recognize and find measures of inscribed angles, and to apply properties of inscribed figures.

  2. Inscribed Angle Inscribed Angle: An angle whose vertex lies on a circle and whose sides are chords of the circle (or one side tangent to the circle). Examples: 3 1 2 4

  3. Intercepted Arc Intercepted Arc: An angle intercepts an arc if and only if : 1. The endpoints of the arc lie on the angle. 2. All points of the arc, except the endpoints, are in the interior of the angle. 3. Each side of the angle contains an endpoint of the arc. Think of Pac Man, his mouth opening is the intercepted arc.

  4. Inscribed Angle Theorem The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. Y Inscribed Angle 110 55 Z Intercepted Arc

  5. Example Name an inscribed angle? Name an arc intercepted by angle BAC. If m BPC = 42, find the m BAC. What is the measure of arc BC?

  6. Another Example • Find X

  7. And Another Example • If the measure of arc ST=68 Find the measure of angle 1 S 1 and angle 2. Note: Q is not the center of the circle. R Q T 2 P

  8. One Last Example… Arc KL is 40 K L Arc LP = 100 P N Arc NP is 60 Find all the angle measures. 6 1 2 3 M 4 5

  9. Theorem   If two inscribed angles of a circle intercept congruent arcs (or the same arc), then the angles are congruent.

  10. Theorem: An angle inscribed in a semicircle is a right angle. P 180 90 S R

  11. Example Using Theorem If the measure of angle HTC = 52°, find the measure of angle CEH. Why? Which angle do you think is 90 °?

  12. Example Using Theorem • If m 4 = 7x +3 and m 5 = 7x +3 and 6 m 1 = 5x. Find x and all the measures of the other angles. • 5 • 3 1 2

  13. Theorem: Inscribed Quadrilaterals If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary. mB + mD = 180  mA+ mC = 180 

  14. Last One…. Make up an example where one angle is given and the opposite angle is unknown. Remember: mD + mB = 180  mA + mC = 180 

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